On stability issues of the HEOM method

TL;DR

This paper investigates the stability issues of the Hierarchical Equations of Motion (HEOM) method in simulating open quantum systems, highlighting how hierarchy truncation can lead to uncontrollable errors especially under strong coupling and reservoir memory effects.

Contribution

The study identifies the causes of critical errors in HEOM due to hierarchy truncation, providing insights into its limitations in certain regimes of open quantum system simulations.

Findings

Truncating HEOM can cause significant errors in strong coupling regimes.

Errors are exacerbated when reservoir memory time is substantial.

Exact results for pure decoherence help diagnose truncation issues.

Abstract

The Hierarchical Equations of Motion (HEOM) method has become one of the cornerstones in the simulation of open quantum systems and their dynamics. It is commonly referred to as a non-perturbative method. Yet, there are certain instances, where the necessary truncation of the hierarchy of auxiliary density operators seems to introduce errors which are not fully controllable. We investigate the nature and causes of this type of critical error both in the case of pure decoherence, where exact results are available for comparison, and in the spin-boson system, a full system-reservoir model. We find that truncating the hierarchy to any finite size can be problematic for strong coupling to a dissipative reservoir, in particular when combined with an appreciable reservoir memory time.